Theorem txprel 30646

Description: A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
txprel	|- Rel ( A (x) B )

Proof of Theorem txprel

Step	Hyp	Ref	Expression
1		txpss3v 30645	. . 3 |- ( A (x) B ) C_ ( _V X. ( _V X. _V ) )
2		xpss 4941	. . 3 |- ( _V X. ( _V X. _V ) ) C_ ( _V X. _V )
3	1, 2	sstri 3441	. 2 |- ( A (x) B ) C_ ( _V X. _V )
4		df-rel 4841	. 2 |- ( Rel ( A (x) B ) <-> ( A (x) B ) C_ ( _V X. _V ) )
5	3, 4	mpbir 213	1 |- Rel ( A (x) B )

Syntax hints:   _Vcvv 3045   C_ wss 3404   X. cxp 4832   Rel wrel 4839   (x) ctxp 30596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-res 4846  df-txp 30620

This theorem is referenced by:  pprodss4v  30651